MULTIPLICATION MODULES WHOSE ENDOMORPHISM RINGS ARE INTEGRAL DOMAINS

In this paper, several properties of endomorphism rings of modules are investigated. A multiplication module M over a commutative ring R induces a commutative ring M* of endomorphisms of M and hence the relation between the prime (maximal) submodules of M and the prime (maximal) ideals of M* can be found. In particular, two classes of ideals of M* are discussed in this paper: one is of the form G(M*)(M, N) = {f is an element of M* | f(M) subset of N} and the other is of the form G(M*) (N, 0) = {f is an element of M* | f(N) = 0} for a submodule N of M.